JEE Mains · Maths · STD 11 - 8. sequence and series
If \(x=\sum \limits_{n=0}^{\infty} a^{n}, y=\sum\limits_{n=0}^{\infty} b^{n}, z=\sum\limits_{n=0}^{\infty} c^{n}\), where \(a , b , c\) are in \(A.P.\) and \(|a| < 1,|b| < 1,|c| < 1\), \(abc \neq 0\), then
- A \(x, y, z\) are in \(A.P.\)
- B \(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\) are in \(A.P.\)
- C \(x, y, z\) are in \(G.P.\)
- D \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1-(a+b+c)\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\) are in \(A.P.\)
Step-by-step Solution
Detailed explanation
\(x =1+ a + a ^{2}=\ldots \ldots \ldots .\) \(x=\frac{1}{1-a} \Rightarrow a=1-\frac{1}{x}\) \(y=\frac{1}{1-b} \Rightarrow b=1-\frac{1}{y}\) \(z=\frac{1}{1-c} \Rightarrow c=1-\frac{1}{z}\) \(a , b , c\) are in \(A.P.\) \(\Rightarrow 1-\frac{1}{x}, 1-\frac{1}{y}, 1-\frac{1}{z}\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The tangent to the parabola \(y^2 = 4x\) at the point where it intersects the circle \(x^2 + y^2 = 5\) in the first quadrant, passes through the pointJEE Mains 2019 Hard
- The number of integers, greater than \(7000\) that can be formed, using the digits \(3,5,6,7,8\) without repetition, isJEE Mains 2023 Medium
- Let \(\overrightarrow{ a }=2 \hat{ i }-3 \hat{ j }+4 \hat{ k }\) and \(\overrightarrow{ b }=7 \hat{ i }+\hat{ j }-6 \hat{ k }\) If \(\overrightarrow{ r } \times \overrightarrow{ a }=\overrightarrow{ r } \times \overrightarrow{ b }, \overrightarrow{ r } \cdot(\hat{ i }+2 \hat{ j }+\hat{ k })=-3,\) then \(\overrightarrow{ r } \cdot(2 \hat{ i }-3 \hat{ j }+\hat{ k })\) is equal toJEE Mains 2021 Hard
- Let the line \(x+y=1\) meet the axes of \(x\) and \(y\) at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines \(O B\) and \(A B\), respectively. If the area of the triangle \(A M N\) is \(\frac{4}{9}\) of the area of the triangle \(O A B\) and AN : NB \(=\lambda: 1\), then the sum of all possible value(s) of is \(\lambda\) :JEE Mains 2025 Hard
- Let \( \alpha, \beta \in \mathbb{R} \) be such that the function
\( f(x)=\begin{cases}2\alpha(x^{2}-2)+2\beta x&,x<1\\ (\alpha+3)x+(\alpha-\beta)&,x\ge1\end{cases} \)
be differentiable at all \( x \in \mathbb{R} \). Then \( 34(\alpha+\beta) \) is equal toJEE Mains 2026 Hard - If \(\mathrm{I}=\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} \mathrm{~d} x\), then \(\int_0^{21} \frac{x \sin x \cos x}{\sin ^4 x+\cos ^4 x} \mathrm{~d} x\) equals :JEE Mains 2025 Hard
More PYQs from JEE Mains
- \(\lim _{n \rightarrow \infty}\left[\frac{1}{n}+\frac{n}{(n+1)^{2}}+\frac{n}{(n+2)^{2}}+\ldots \ldots .+\frac{n}{(2 n-1)^{2}}\right]\) is equal to ...... .JEE Mains 2021 Hard
- If the sum of the first 20 terms of the series
\(\frac{4.1}{4+3.1^2+1^4}+\frac{4.2}{4+3.2^2+2^4}+\frac{4.3}{4+3.3^2+3^4}+\frac{4.4}{4+3.4^2+4^4}+\ldots\)
is \(\frac{m}{n}\), where \(m\) and \(n\) are coprime, then \(m+n\) is equal to :-JEE Mains 2025 Medium - Let \(A=\{z \in C:|z-2-i|=3\}\), \(B=\{z \in C: \operatorname{Re}(z-i z)=2\}\) and \(S=A \cap B\). Then \(\sum_{z \in S}|z|^2\) is equal to ________ .JEE Mains 2025 Medium
- A \(10\, inches\) long pencil \(\mathrm{AB}\) with mid point \(\mathrm{C}\) and a small eraser \(\mathrm{P}\) are placed on the horizontal top of a table such that \(\mathrm{PC}=\sqrt{5}\) inches and \(\angle \mathrm{PCB}=\tan ^{-1}(2)\). The acute angle through which the pencil must be rotated about \(\mathrm{C}\) so that the perpendicular distance between eraser and pencil becomes exactly \(1\, inch\) is:
JEE Mains 2021 Hard - Let \(\mathrm{E}\) be an ellipse whose axes are parallel to the co-ordinates axes, having its center at \((3,-4)\), one focus at \((4,-4)\) and one vertex at \((5,-4) .\) If \(m x-y=4, m\,>\,0\) is a tangent to the ellipse \(\mathrm{E}\), then the value of \(5 \mathrm{~m}^{2}\) is equal to \(.....\)JEE Mains 2021 Hard
- If the coefficients of \(x^{7}\) in \(\left(x^{2}+\frac{1}{b x}\right)^{11}\) and \(x^{-7}\) in \(\left(x-\frac{1}{b x^{2}}\right)^{11}, b \neq 0\), are equal, then the value of \(b\) is equal to:JEE Mains 2021 Hard